Stable random variables with complex stability index, I

The present paper is the first part of a work on stable distributions with a complex stability index. We construct complex-valued random variables (r.v.'s) satisfying the usual stability condition but for a complex parameter $\alpha$ such that $|\alpha-1|<1$. We find the characteristic functions (ch.f.'s) of the r.v.'s thus obtained and prove that their distributions are infinitely divisible. It is also shown that the stability condition characterizes this class of stable r.v.'s.